A technique is developed for obtaining the phase space density distribution within the acceptance of a synchrotron if the distribution of the emittance is known. It is shown that the matching of an injector emittance to a synchrotron acceptance depends not only on the phase space boundary but also on the phase space density distribution. The problem of maximizing the charge accepted by a synchrotron with respect to the initial accelerating voltage is considered. Two cases are treated analytically which, while highly idealized, illustrate the general method and give upper and lower bounds on the voltage. A numerical computation is carried out to obtain the optimum synchrotron voltage at injection for a practical synchrotron injection system.